4
votes
1answer
130 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
0
votes
2answers
101 views

General question about undergrad math classes? [closed]

I'm currently in Real Analysis 2, and while I'm doing pretty well, it's just so much time and effort to finish all the problem sets and do well on the exams. I only have a few math classes left for ...
0
votes
2answers
49 views

An ABC soft question about epsilon-delta argument

Someone told me that some textbooks present epsilon-delta argument somewhat misleadingly. For example, consider the simplest one: the convergence of the sequence $(1/n)_{1}^{\infty}$ to $0$. These ...
2
votes
2answers
115 views

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
2
votes
0answers
53 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
1
vote
1answer
125 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. ...
1
vote
0answers
103 views

Missing connection in real analysis?

I have long thought of real analysis as the consequences of mapping numbers to real numbers, thereby creating infinite sequences and series. From there, I can get to fairly elementary applications ...
26
votes
8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
0answers
73 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
0
votes
0answers
24 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
8
votes
3answers
531 views

What level of rigour is expected in Real Analysis?

I fail to find a duplicate. I am wondering what level of rigour is needed in a typical undergraduate course in Real Analysis. To clarify my question, I provide an exercise from Rudin and my proposed ...
42
votes
7answers
5k views

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
2
votes
3answers
96 views

Does $f(x)\,dx$ denote multiplication of $f(x)$ by $dx$? [duplicate]

In the integral form $\int \! f(x) \, \mathrm{d}x$ does $f(x)\,\mathrm{d}x$ can be seen as a multiplication of $f(x)$ and $\mathrm{d}x$?
4
votes
3answers
438 views

I am failing at Analysis again. [closed]

I am stuck at this very hard course again. I don't want to put blames but if you were to ask me, what is going wrong I would say: The professor teaches too bluntly The book is too hard to ...
5
votes
2answers
203 views

How to think of zeros of the derivative of a holomorphic funcion?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...
2
votes
3answers
94 views

Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction.

Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction. I've been thinking how to prove the above statement not using contradiction. My idea for a ...
14
votes
2answers
649 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
3
votes
1answer
112 views

What is the best way to tell people what Analysis is about?

What is the best way to tell people what Analysis is about? I am currently taking Analysis course. However, I am really having a big difficulty explaining to people what Mathematical Analysis is ...
1
vote
1answer
79 views

Prove that $\int_{-\pi}^{\pi}$ $\frac{d\theta}{1+\sin^2\theta}$ = $\pi\sqrt{2}$ using the method of Residues

Prove that $$\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta} = \pi\sqrt{2}$$ using the method of Residues How do I do this? I know I need it from $0$ to $2\pi$ but I don't know how to modify it!! ...
0
votes
1answer
33 views

How can I prove that $_{max}|Az^{n}+b|$ =$ |A|$ + $|B|$ when |Z| $\leq$ 1?

How can I prove that $_{max}$$|Az^{n}+B|$ = $|A| + |B|$ when $|Z|$ $\leq$ 1? Remember that Z is a complex number which is why I had to include the magnitude.
1
vote
0answers
41 views

Uses of the Heine-Cantor theorem?

Heine-Cantor Theorem: Let $[a,b]$ be a compact interval, $f:[a,b]\to R$ be a continuous function. Then $f$ is uniformly continuous, i.e. $\forall \epsilon>0 , \exists \delta>0$ such that ...
2
votes
1answer
194 views

Is RCA-Rudin one of the worst textbooks? [closed]

Someone told me that "Real and complex analysis - rudin" is actually rated a bad textbook among researchers, since it gives no motivation. Is it true? I agree that this text provides less ...
2
votes
0answers
60 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...
3
votes
1answer
96 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
16
votes
11answers
902 views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
2
votes
0answers
30 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
3
votes
3answers
139 views

How to know if I'm doing real analysis correctly?

I've just started a second year course in real analysis. This is my first proof-oriented course. Last year, our maths curriculum was introductory tertiary calculus and algebra. When I practised ...
2
votes
1answer
44 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
1
vote
1answer
53 views

Can Bohr-Mollerup Theorem be extended to complex plane?

There is a famous theorem says about Gamma function: Bohr-Mollerup Theorem Let $f:(0,\infty)\rightarrow \mathbb{R}^+$ be a function satisfying below (i) $f(x+1)=xf(x), \forall x\in ...
5
votes
2answers
92 views

Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
2
votes
0answers
121 views

recommendation on studying real-analysis

I hope someone who had studied RCA would answer my question. I bought Rudin-Real and Complex analysis for self-study. (Please don't mind that it is a self-study. I don't care at all whether a text is ...
2
votes
1answer
91 views

Derivative existence theorem

Has anyone here heard of the Derivative existence theorem? Derivative existence theorem: For $f$ defined on some interval including $a$, $f$ is differentiable at $a$ if and only if there ...
1
vote
2answers
123 views

Infinite sum with 0 terms: comparison to infinite product

Depend on what text you read, an infinite product with an infinite number of terms that are 0 is either divergent, or diverge to 0. Even though, obviously, the partial product is still a convergence ...
11
votes
1answer
236 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
21
votes
3answers
773 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
3
votes
1answer
402 views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
3
votes
2answers
263 views

How does one get better at real analysis proofs?

How does one proceed through a math proof in real analysis? My instructor always says make a diagram, but I am not a visual learner. It seems that whenever I write out the definition of an assumption, ...
1
vote
0answers
48 views

A Question Related to the Divergence Test for Series

Suppose $\lbrace x_i\rbrace$ is a sequence of real numbers. If $\lim x_i=0$ then for every $\varepsilon>0$ there exists an $N$ such that $$n\ge N\Rightarrow \vert ...
0
votes
0answers
49 views

Between any rational and $\sqrt{2}$ is another rational [duplicate]

I'm self-studying analysis from Rudin, and in there is this proof. I understand the proof after reading it, but how would a mathematician come up with this or approach doing a proof like this? It ...
1
vote
1answer
94 views

Analysis for Engineers: Where Do You Start?

Having taken none of the prerequisite rigorous treatments of mathematics during my undergrad years, I feel at a disadvantage to the people in my major what do have that analysis/abstract math ...
2
votes
2answers
83 views

Why study difference equation, sequences, etc?

A couple of weeks ago I studied a thorough(yet basic) course in ODE, part of an course in analysis. Shortly after, we moved on to the study of difference equations, which is very much similar to ...
1
vote
2answers
125 views

Why is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
5
votes
2answers
195 views

Is $(-\infty,\infty)$ a closed **interval**?

Note that we are working in the reals, not the extended reals. Do you understand a closed interval as "an interval that is a closed set" or as "an interval that includes both its endpoints"? If the ...
1
vote
1answer
63 views

Is a closed n-dimensional disk compact necessarily compact?

As the title asks, is a closed n-dimensional disk compact necessarily compact? I'm thinking the answer would be no. If you consider the case in $\mathbb{R}^1$ then can you define the radius to be ...
1
vote
0answers
83 views

On Cauchy Sequences

I would consider this a soft question because I am seeking some insight on how to work with Cauchy sequences by using the Cauchy criterion for convergence. To my understanding, the definition is ...
3
votes
2answers
224 views

Question on questions in Spivak's Calculus?

I started reading Spivak's Calculus about a month ago and I'm at the end of chapter two, so this is not really calculus yet. However, I find the problems really difficult and the answer keys are not ...
2
votes
5answers
158 views

What's so special about $e$? [duplicate]

If someone with not much mathematics in his luggage asks me: What is so special about $\pi$? then off course I have an answer. Even if $i$ would be the subject (I allready see him gazing at my ...
0
votes
0answers
47 views

Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
2
votes
0answers
106 views

Is my general approach to proofs acceptable? A general topology example.

Proving: $A$ is closed iff $A = \bar{A}$. "To the right": If $A$ is closed, $ A = \bar A$ If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
5
votes
1answer
673 views

Unexpected Practical Applications of Calculus

Calculus shows up in a lot of places in the world. Specifically, here are three areas where I see it used the most: Optimization problems. Anything involving rates of change (e.g. velocity ...